A nonlinear model for rotationally constrained convection with Ekman pumping

It is a well established result of linear theory that the influence of differing mechanical boundary conditions, i.e., stress-free or no-slip, on the primary instability in rotating convection becomes asymptotically small in the limit of rapid rotation. This is accounted for by the diminishing impact of the viscous stresses exerted within Ekman boundary layers and the associated vertical momentum transport by Ekman pumping. By contrast, in the nonlinear regime recent experiments and supporting simulations are now providing evidence that the efficiency of heat transport remains strongly influenced by Ekman pumping in the rapidly rotating limit. In this paper, a reduced model is developed for the case of low Rossby number convection in a plane layer geometry with no-slip upper and lower boundaries held at fixed temperatures. A complete description of the dynamics requires the existence of three distinct regions within the fluid layer: a geostrophically balanced interior where fluid motions are predominately aligned with the axis of rotation, Ekman boundary layers immediately adjacent to the bounding plates, and thermal wind layers driven by Ekman pumping in between. The reduced model uses a classical Ekman pumping parameterization to alleviate the need for spatially resolving the Ekman boundary layers. Results are presented for both linear stability theory and a special class of nonlinear solutions described by a single horizontal spatial wavenumber. It is shown that Ekman pumping allows for significant enhancement in the heat transport relative to that observed in simulations with stress-free boundaries. Without the intermediate thermal wind layer the nonlinear feedback from Ekman pumping would be able to generate a heat transport that diverges to infinity. This layer arrests this blowup resulting in finite heat transport at a significantly enhanced value.Comment: 38 pages, 14 figure

POROUS MEDIUM CONVECTION AT LARGE RAYLEIGH NUMBER: STUDIES OF COHERENT STRUCTURE, TRANSPORT, AND REDUCED DYNAMICS

Buoyancy-driven convection in fluid-saturated porous media is a key environmental and technological process, with applications ranging from carbon dioxide storage in terrestrial aquifers to the design of compact heat exchangers. Porous medium convection is also a paradigm for forced-dissipative infinite-dimensional dynamical systems, exhibiting spatiotemporally chaotic dynamics if not ``true turbulence. The objective of this dissertation research is to quantitatively characterize the dynamics and heat transport in two-dimensional horizontal and inclined porous medium convection between isothermal plane parallel boundaries at asymptotically large values of the Rayleigh number $Ra$ by investigating the emergent, quasi-coherent flow. This investigation employs a complement of direct numerical simulations (DNS), secondary stability and dynamical systems theory, and variational analysis. The DNS confirm the remarkable tendency for the interior flow to self-organize into closely-spaced columnar plumes at sufficiently large $Ra$ (up to $Ra \simeq 10^5$), with more complex spatiotemporal features being confined to boundary layers near the heated and cooled walls. The relatively simple form of the interior flow motivates investigation of unstable steady and time-periodic convective states at large $Ra$ as a function of the domain aspect ratio $L$. To gain insight into the development of spatiotemporally chaotic convection, the (secondary) stability of these fully nonlinear states to small-amplitude disturbances is investigated using a spatial Floquet analysis. The results indicate that there exist two distinct modes of instability at large $Ra$: a bulk instability mode and a wall instability mode. The former usually is excited by long-wavelength disturbances and is generally much weaker than the latter. DNS, strategically initialized to investigate the fully nonlinear evolution of the most dangerous secondary instability modes, suggest that the (long time) mean inter-plume spacing in statistically-steady porous medium convection results from an interplay between the competing effects of these two types of instability. Upper bound analysis is then employed to investigate the dependence of the heat transport enhancement factor, i.e. the Nusselt number $Nu$, on $Ra$ and $L$. To solve the optimization problems arising from the ``background field upper-bound variational analysis, a novel two-step algorithm in which time is introduced into the formulation is developed. The new algorithm obviates the need for numerical continuation, thereby enabling the best available bounds to be computed up to $Ra\approx 2.65\times 10^4$. A mathematical proof is given to demonstrate that the only steady state to which this numerical algorithm can converge is the required global optimal of the variational problem. Using this algorithm, the dependence of the bounds on $L(Ra)$ is explored, and a ``minimal flow unit is identified. Finally, the upper bound variational methodology is also shown to yield quantitatively useful predictions of $Nu$ and to furnish a functional basis that is naturally adapted to the boundary layer dynamics at large $Ra$

Bounds on heat transport for internally heated convection

Convection of a fluid between parallel plates driven by uniform internal heating is a problem where the asymptotic scaling of the mean vertical convective heat transport ⟨wT⟩ was largely unknown. This thesis proves upper bounds on ⟨wT⟩ with respect to the non-dimensional Rayleigh number R. Here R quantifies the destabilising effect of heating compared to the stabilising effect of diffusion. By the background field method, formulated in terms of quadratic auxiliary functionals, linear convex optimisation problems are constructed whose solutions provide upper bounds on ⟨wT⟩. The numerical optimisation carried out with semidefinite programming guides the mathematical analysis and subsequent proofs. The quantity ⟨wT⟩ has different physical implications based on the three thermal boundary conditions studied: perfect conductors, an insulating bottom and perfectly conducting top, and poorly conducting boundaries. In the first setup, ⟨wT⟩ quantifies the flux of heat out of the top and bottom. Whereas in the latter two cases, ⟨wT⟩ quantifies the ratio of total heat transport to the mean conductive heat transport. Critical to the proofs is the use of a minimum principle on the temperature. Finally, we also prove bounds in the scenarios of infinite Prandtl numbers and free-slip boundaries.Open Acces

Geostrophic and chimney regimes in rotating horizontal convection with imposed heat flux

Convection in a rotating rectangular basin with differential thermal forcing at one horizontal boundary is examined using laboratory experiments. The experiments have an imposed heat flux boundary condition, are at large values of the flux Rayleigh number ($Ra_{F}\sim O(10^{13}{-}10^{14})$ based on the box length $L$), use water with Prandtl number $Pr\approx 4$ and have a small depth to length aspect ratio. The results show the conditions for transition from non-rotating horizontal convection governed by an inertial–buoyancy balance in the thermal boundary layer, to circulation governed by geostrophic flow in the boundary layer. The geostrophic balance constrains mean flow and reduces the heat transport as Nusselt number $Nu\sim (Ra_{F}Ro)^{1/6}$, where $Ro=B^{1/2}/f^{3/2}L$ is the convective Rossby number, $B$ is the imposed buoyancy flux and $f$ is the Coriolis parameter. Thus flow in the geostrophic boundary layer regime is governed by the relative roles of horizontal convective accelerations and Coriolis accelerations, or buoyancy and rotation, in the boundary layer. Experimental evidence suggests that for more rapid rotation there is another transition to a regime in which the momentum budget is dominated by fluctuating vertical accelerations in a region of vortical plumes, which we refer to as a ‘chimney’ following related discussion of regions of deep convection in the ocean. Coupling of the chimney convection in the region of destabilising boundary flux to the diffusive boundary layer of horizontal convection in the region of stabilising boundary flux gives heat transport independent of rotation in this ‘inertial chimney’ regime, and the new scaling $Nu\sim Ra_{F}^{1/4}$. Scaling analysis predicts the transition conditions observed in the experiments, as well as a further ‘geostrophic chimney’ regime in which the vertical plumes are controlled by local geostrophy. When Ro<10^{-1}, the convection is also observed to produce a set of large basin-scale gyres at all depths in the time-averaged flow.</jats:p